Let us first recall the theoretical features from reference frame
transformations in general, and spatial translations in particular,
as we understood in classical physics. The group of admissible
reference frame transformations is a Lie group that is considered
as the relativity symmetry for the theory. The perspective of it
as a symmetry of our physical space-time is used to be considered
fundamental. In fact, the relativity symmetry and the model of
space-time, or spacetime, are really tied together. For example,
having Lorentz symmetry requires the Minkowski spacetime as
the `correct' model whereas its Galilean limit is to be matched
to the Newtonian space-time. Actually, the Newtonian space-time
and the Minkowski spacetime can be theoretically constructed
each as a coset space of the Galilean and Poincaré groups,
respectively (see Ref.[1]} for an explicit illustration).
Physics is, however, really about dynamics; and dynamics is
to be considered on states of physical systems, which does not
include space and time themselves in theories not containing
gravitational dynamics. In this paper, we focus only on simple
quantum mechanics, or the quantum version of Newtonian
mechanics. The space of all physical states for a Newtonian
particle can also be constructed as a coset space of the Galilean
group [1]}. This particle phase space splits into the
configuration/position space and the momentum space. The
single particle configuration space as the space of all possible
positions of the particle is what constitutes a model for the
physical space. The phase space is naturally a symplectic
manifold, which can be seen as a geometric space with symmetry
transformations naturally to be formulated as Hamiltonian flows
[1]}. In fact, the Poisson bracket structure is essentially
dictated by the Galilean group as a Lie group [4]}. Practical
dynamics corresponds to the identification of the physical
Hamiltonian, the energy observable, as among all possible
generic Hamiltonian functions. Every generator of the
Galilean group corresponds to an observable whose
Hamiltonian vector field represents the abstract generator.
It generates a one-parameter group of Hamiltonian flows on
the phase space preserving the symplectic structure, as well
as an automorphism flow on the observable algebra preserving
the Poisson bracket [1]}. The observable algebra itself
can be seen as the representation of the universal enveloping
algebra, the group algebra, or some generalization of it, fixed
by the representations of the generators. Here the observable
algebra represented as functions on the phase space with the
usual product is commutative.
